I have now drawn the 10x10 arrangement and have written the actual properties as read from my sketch in this table:
From the results gathered by actually drawing the arrangement I have deduced that the rules that I obtained from my original set of results are in fact correct for the square arrangements that I have drawn so far, and they appear to be able to hold true when tested with larger arrangements. I have therefore decided to move onto the next stage of my investigation.
Stage 2 – Rectangular arrangements
For the next stage of my investigation, I will investigate the rules for spacers when used with rectangular arrangements. These rules will apply to any arrangement of squares, as long as they are in a rectangle. Therefore I do not need to draw out a table like the squares section.
The rule for the number of squares in the squares section, xy, will apply to rectangles as well. The rule for the L – shaped spacer that I have obtained from the squares section will also apply to rectangular shapes, as squares and rectangles both have four corners. The rule for the T shaped spacers is 2(x-1) + 2(y-1). This is because there is always one less T shaped spacer than the number of squares in an edge. So, I need to take one away from each edge (this is done in the brackets). Then a need to multiply each the result by two, as there is two of each edge. Then I need to add the results together, to get the total number of T shaped spacers. The rule for the + shaped spacer is (x-1)(y-1). This is because as the + shaped spacers do not reach to the edge of the rectangle, one needs to be taken of each edge. Then the results of this need to be multiplied together to get the total.
The tests I made showed that the predictions that I have made are in fact correct, enabling me to conclude this section of my work without further investigation.
Stage 3 – Triangular arrangements
To investigate triangular arrangements I need to invent three new types of spacers. The new spacers are:
^ Spacer K Spacer * Spacer
I began by drawing a sequence of triangles on a separate sheet of paper. (See sheet T1). Here is a table showing he information for the first five arrangements:
The rule for the ^ shaped spacer will always be three, because there are always three corners on a triangle. The rule for the K shaped spacers is 3(n-1). The amount of K shaped spacers in one side of the triangle is always one less than the pattern number. So at some point I shall need to multiply by three. But before I can do that I need to subtract one from the pattern number, hence the n-1 in brackets. Then I need to multiply this by three, as there are three sides in a triangle.
When investigating the rule for the * shaped spacers, I came across a problem. I tried to use quadrilateral equations to find a rule, but this didn’t work. So I decided to approach the problem from a different perspective. Firstly, I doubled the amount of * spacers in a triangle, thus forming a rectangle. (See sheet T2). Now I could apply the rule I used to find the amount of squares in a rectangle, which is xy. I saw that x is always two less than the pattern number (n), and y is always one less. So I made the rule (n-2) x (n-1). But this was always double the amount of the * shaped spacers. Dividing the rule by two easily solved this. So I came out with the rule
(n-2) x (n-1)
2
I simplified this to n - 3n + 1
2
I now feel that I have fulfilled my original coursework assignment and do not have to investigate further. I have encountered some problems, which I overcame.
Name: Sam Koprowski
Candidate Number: 7393
School: George Ward
Centre Number: 66633